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Sociological Methods

DOI: 10.1177/0049124106292364 2007; 35; 542 Sociological Methods Research

Bobby L. Jones and Daniel S. Nagin Procedure for Estimating Them

Advances in Group-Based Trajectory Modeling and an SAS

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Advances in Group-BasedTrajectory Modeling and

an SAS Procedurefor Estimating Them

Bobby L. JonesDaniel S. NaginCarnegie Mellon, Pittsburgh, PA

This article is a follow-up to Jones, Nagin, and Roeder (2001), which described

an SAS procedure for estimating group-based trajectory models. Group-

based trajectory is a specialized application of finite mixture modeling and is

designed to identify clusters of individuals following similar progressions of

some behavior or outcome over age or time. This article has two purposes.

One is to summarize extensions of the methodology and of the SAS proce-

dure that have been developed since Jones et al. The other is to illustrate

how group-based trajectory modeling lends itself to presentation of findings

in the form of easily understood graphical and tabular data summaries.

Keywords: developmental trajectories; Proc Traj; finite mixture models

his article is a follow-up to Jones, Nagin, and Roeder (2001), which

described an SAS procedure for estimating group-based trajectory

models. The Proc Traj software can be downloaded from www.andrew

.cmu.edu/user/bjones/index.htm free of charge. Also available for down-

load at the Web site are documentation and a number of supporting SAS

macros described in this article and the Jones et al. article.

Group-based trajectory models are designed to identify clusters of indi-

viduals following similar progressions of some behavior or outcome over

age or time. Psychologists call such progressions developmental trajec-

tories. We borrow this term to describe the course of any phenomenon,

whether behavioral, biological, or physical. Charting and understanding

Sociological Methods

& Research

Volume 35 Number 4

May 2007 542-571

� 2007 Sage Publications

10.1177/0049124106292364


hosted at

http://online.sagepub.com

Authors’ Note: The research was supported by generous financial support from the National

Science Foundation (SES-9911370) and the National Institute of Mental Health (RO1

MH65611-01A2).

T

542




developmental trajectories is among the most fundamental and empiri-

cally important research topics in the social and behavioral sciences and

medicine.

As developed in Nagin (1999, 2005), the group-based trajectory model is

a specialized application of finite mixture modeling. Piquero (2004) reports

that more than 60 articles have been published that use group-based trajec-

tory modeling. This article has two purposes. One is to summarize extensions

of the methodology and of the SAS procedure that have been developed

since Jones et al. (2001). The other is to illustrate how group-based trajectory

modeling lends itself to presentation of findings in the form of easily under-

stood graphical and tabular data summaries. Data summaries of this form

have the great advantage of being accessible to nontechnical audiences and

quickly comprehensible to audiences that are technically sophisticated.

Brief Overview of the Statistical Model

Suppose Yi = y;i1yi2; yi3; . . . ; y1T

� �represents the longitudinal sequence

of measurements on an individual i over T periods, and PðYiÞ denotes the

probability of Yi. The group-based trajectory model assumes that the popula-

tion is composed of a mixture of J underlying trajectory groups such that

PðYiÞ =Xj

πjPjðYiÞ;

where PjðYiÞ is the probability of Yi given membership in group j, and

πj is the probability of group j. The basic model also assumes that condi-

tional on membership in group j, the random variables, yit, t= 1; 2; . . . ; T ,

are independent. Thus, PjðYiÞ= QTpjtðyitÞ.

The group membership probabilities, πj, j= 1; . . . ; J , are not estimated

directly but instead are estimated by a multinomial logit function:

πj = eθj=XJ

1eθj ; ð1Þ

where θ1 is normalized to zero. Estimation of πj in this fashion ensures

that each such probability properly falls between 0 and 1.

The form of pjtðyitÞ is selected to conform to the type of data under

analysis. If yit is a count variable, then pjtðyitÞ is assumed to be the Poisson

distribution or even more general zero-inflated Poisson (Lambert 1993).

For psychometric scale data, pjtðyitÞ is assumed to follow the censored nor-

mal distribution to accommodate the possibility of clustering at the scale

Jones, Nagin / Group-Based Trajectory Modeling 543




minimum and maximum. For binary data, pjtðyitÞis assumed to follow the

binary logit distribution.

As in most hierarchical and latent curve models, a polynomial relation-

ship is used to model the link between age and behavior. The Proc Traj

software allows estimation of up to a fourth-order polynomial. For the

Poisson-based model, it is assumed that

lnðλjitÞ=βj0 +β

j1Ageit +β

j2Age

2it +β

j3Age

3it +β

j4Age

4it; ð2Þ

where λjit is the expected number of occurrences of the event of interest

(e.g., convictions) of subject i at time t, given membership in group j, and

Age it is subject i’s age at time t. The model’s coefficients—βj0;β

j1;β

j2;β

j3,

and βj4—determine the shape of the trajectory. Because this vector of para-

meters, which we denote by βj, is trajectory group specific, its values can

vary freely across the j groups.

For the censored normal model, the linkage between age and behavior

is established via a latent variable, y∗jit , that can be thought of as measuring

the potential for engaging in the behavior of interest, say physical aggres-

sion. Again, up to a fourth-order polynomial relationship is assumed

between y∗jit and age:

y∗jit =β

j0 +β

j1Ageit +β

j2Age

2it +β

j3Age

3it +β

j4Age

4it + εit; ð3Þ

where εit is a disturbance assumed to be normally distributed with a zero

mean and a constant standard deviation σ.

In the case of binary data, it is assumed that conditional on membership

in group j, pjtðyitÞ follows the binary logit distribution:

pjtðyitÞ= eβj

0+β

j

1Ageit +β

j

2Age2

it+β

j

3Age3

it+β

j

4Age4

it

1+ eβj

0+β

j

1Ageit +β

j

2Age2

it+β

j

3Age3

it+β

j

4tAge4

it

: ð4Þ

Data Sources

Two major longitudinal studies are used to illustrate the modeling

extension demonstrated in this article.

Cambridge Study of Delinquent Development (the London study). This

study tracked a sample of 411 British males from a working-class area of

London. Data collection began in 1961-1962, when most of the boys were

age 8. Criminal involvement is measured by the number of convictions for

criminal offenses and is available for all individuals in the sample through

544 Sociological Methods & Research




age 32, with the exception of the eight individuals who died prior to this

age. Between ages 10 and 32, a wealth of data was assembled on each indi-

vidual’s psychological makeup; family circumstances, including parental

behaviors; and performance in school and work. The modeling demonstra-

tions are based on the 403 individuals who survived to age 32. For a com-

plete discussion of the data set, see Farrington (1990).

Montreal Longitudinal-Experimental Study of Boys (the Montreal study).

The 1,037 participants in this study were part of a longitudinal study started

in the spring of 1984. All teachers of kindergarten classes in the 53 schools

of the lowest socioeconomic areas in Montr�eal (Canada) were asked to rate

the behavior of each boy in their classroom. To control for cultural effects,

the boys were included in the longitudinal study only if both their biological

parents were born in Canada and their biological parents’ mother tongue

was French. Assessments were made at age 6 and annually from ages 10 to

17. Wide-ranging measurements of social and psychological function were

made based on assessments by parents, teachers, and peers; self-reports by

the boy himself; and administrative records from schools and the juvenile

court. See Tremblay et al. (1987) for further details on this study.

Modeling and Software Extensions

Maximum likelihood is used for the estimation of the model para-

meters. The maximization is performed using a general quasi-Newton pro-

cedure (Dennis, Gay, and Welsch 1981; Dennis and Mei 1979). Because

the procedure may find only a local maximum, multiple starting points are

specified so that there is reasonable assurance that the global maximum is

located. The variance-covariance matrix for the parameter estimates is

obtained from the inverse observed information matrix evaluated at the

maximum likelihood parameter estimates. Wald tests for parameters equal

to zero are performed for the trajectory parameters and generalized logit

parameters. The Bayesian information criterion (BIC) is relied on for

model selection as described in Jones et al. (2001).

Extension 1: Confidence Intervals on Trajectories

and Group Membership Probabilities

The direct products of the model estimation are the estimates of the

vectors that describe each group j’s trajectory, β j, and the vector that

is composed of the parameters that determine the group membership





probabilities, θ. Also estimated is the variance-covariance matrix for the

parameter estimates. However, the quantities of actual interest to the user

are the group membership probabilities, as specified by equation (1), and

the trajectories, as specified by equation (2), (3), or (4). Because these

quantities are nonlinear transformations of the parameter estimates, esti-

mation of their standard errors and attendant confidence intervals requires

more complex calculations.

One approach to estimating the standard errors of the desired quantities

is the bootstrap method (Efron 1979). We did not choose this alternative

because the resampling requirements would make model estimation time

prohibitive for most users. The other alternative is to approximate the stan-

dard error by a first-order Taylor series expansion (Greene 1995). We used

this alternative because it is easily implemented and has no material

impact on estimation time.

We illustrate this extension with an application to the London data in

which we estimate a four-group Poisson-based model. Figure 1 shows the

point estimates of the trajectories of λjt and their accompanying 95 percent

confidence intervals. The fact that the confidence intervals at most overlap

for only relatively short periods indicates that the trajectories are distinct.

One group is composed of individuals with a miniscule rate of offending

over the entire observation period. This trajectory of negligible offending,

which we call the nonoffending group, was modeled by a zero-order (i.e.,

constant) trajectory in age. Such a zero-order trajectory could also capture

the trajectory of a constant high-rate trajectory. Probability of membership

in this group is estimated at 69.5 percent, with a 95 percent confidence inter-

val (CI) of 63.3 to 75.7 percent. The three remaining trajectory groups were

specified to follow a quadratic function of age. We refer to Group 2 as the

‘‘adolescent-limited’’ group because following a period of active offending

during adolescence, the predicted λjt is negligible after age 20. The estimated

probability of membership in this group is 12.4 percent (95 percent CI:

7.1 percent, 17.7 percent). Group 4 is referred to as the high-chronic group

because of its high offense rate during adolescence and early adulthood and

its nonnegligible rate thereafter. This trajectory characterized the behavior of

an estimated 5.9 percent of the population (95 percent CI: 3.4 percent,

8.4 percent). The final quadratic trajectory, Group 3, is referred to as the low

chronics because members of this group offended at a nonnegligible rate

throughout the observation period. The size of this group is estimated at 12.2

percent of the sampled population (95 percent CI: 7.9 percent, 16.5 percent).

The following statements fit this model and produce the graph shown

in Figure 1:





proc traj data=london_data out=london_out outstat=crimstat

outplot=crimplot ci95m;

var y1-y11;

indep t1-t11;

model zip;

ngroups 4;

start -5 -.5 0 0 0 0 0 .5 0 0 70 10 10 10;

order 0 2 2 2;

%trajplotnew (crimplot,crimstat, ‘Conviction Rate vs. Age’,

‘Four Group-Poisson Model’,’Lambda’,’Age/10’)

10 15 20 25 30

0.0

0.5

1.0

1.5

2.0

2.5

3.0

Age

An

nu

al C

on

vict

ion

Rat

e

5.9 percent 12.2 percent 12.4 percent 69.5 percent

Figure 1

Annual Conviction Rate Versus Age: Four-Group Poisson Model





Extension 2: Dual-Trajectory Models

The dual model was designed to analyze the developmental course of

two distinct but related outcomes (Nagin and Tremblay 2001). The model

can be used to analyze connections between the developmental trajectories

of two outcomes that are evolving contemporaneously (e.g., depression

and alcohol use) or that evolve over different time periods (e.g., prosocial

behavior in childhood and school achievement in adolescence).

The three key outputs of the dual model are as follows: (1) the trajectory

groups for both measurement series, (2) the probability of membership in

each identified trajectory group, and (3) conditional probabilities linking

membership across the trajectory groups of the two respective behaviors.

Loeber (1991) has argued that covert behaviors in childhood, such as

opposition, are linked to another form of covert behavior in adolescence,

property delinquency. We thus illustrated the dual model with an analysis

of the linkage of opposition from ages 6 to 13 with property delinquency

from ages 13 to 17. The model is estimated with data from the Montreal-

based longitudinal study.

Figure 2 displays the form of the trajectories identified for these two

behaviors. Panel A shows the trajectories of opposition from ages 6 to 13,

which were a product of the censored normal model. One trajectory starts

off low at age 6 and declines steadily thereafter. The second trajectory

starts off at a modest level of opposition at age 6, rises slightly until age

10, and then begins a gradual decline. The third trajectory starts off high

and remains high over the age period. These trajectories of childhood

opposition were estimated to account for 32.0, 45.2, and 22.9 percent of

the population, respectively.

Panel B shows the trajectories for property delinquency, which were

estimated using the Poisson-based model. The two largest trajectory

groups, which were estimated to account for a total of nearly 72 percent of

the population, are characterized by low and slightly declining rates of

property delinquency. The third trajectory group follows a pattern of rising

property delinquency over most of the measurement period, whereas the

fourth group remains high over the entire period. Groups 3 and 4 were esti-

mated to account for 19.2 and 8.9 percent of the population, respectively.

Figure 3 reports the key innovation of the joint trajectory model—three

alternative representations of the linkage between opposition and property

delinquency. One is the probability of membership in each of the property

delinquency trajectories, conditional on membership in each of the opposi-

tion trajectory groups. These probabilities, which are reported in Panel A,





(B)

0

2

4

6

8

10

12

14

13 14 15 16 17Age

Rat

e

Low 1 (36.0%) Low 2 (35.9%)Rising (19.2%) Chronic (8.9%)

(A)

0

1

2

3

4

5

6

7

76 98 10 11 12 13

Age

Op

po

siti

on

Low (31.9%) Moderate (45.2%) High (22.9%)

Figure 2

(A) Trajectories of Opposition From Ages 6 to 13 and

(B) Trajectories of Property Delinquency From Ages 13 to 17





can be interpreted as the probability of transitioning from each childhood

opposition trajectory to each of the property delinquency trajectories.

Because the probabilities are conditional on membership in a given opposi-

tion trajectory group, each column of probabilities in Panel A sums to 1.

Panel B reports the reverse set of conditional probabilities: the probability of

membership in each of the opposition trajectories conditional on membership

1.0

0.8

0.6

0.4

0.2

0.0Low 1 Low 2

(A) Probability of delinquency group conditional on opposition group

Rising Chronic

Low opposition High oppositionModerate opposition

1.00.80.60.40.20.0

(B) Probability of oppositional group conditional on delinquency group

Low Opposition Moderate Opposition High Opposition

Low 1 Low 2 Rising Chronic

1.00.80.60.40.20.0

(C) Joint probability of opposition group and delinquency group

Low opposition High oppositionModerate opposition

Low 1 Low 2 Rising Chronic

Figure 3

The Relationship of Opposition (Ages 6-13)

With Property Delinquency (Ages 13-17)





in each of the property delinquency trajectory groups. In this panel, each row

of probabilities sums to 1. These probabilities measure the composition of

each adolescent trajectory group in terms of the childhood trajectory of ori-

gin. The third form of representation, reported in Panel C, is the joint prob-

ability of membership in a specific property delinquency trajectory and a

specific trajectory of the opposition trajectory group. This panel enumerates

the probabilities of all the possible combinations of opposition and property

delinquency trajectory groups. Thus, the 12 joint probabilities sum to 1.

However represented, the results show a strong relationship between

the developmental trajectories for these two behaviors. Panel A shows that

the boys who were least oppositional from ages 6 to 13 were least likely

to be members of the two higher trajectories of property delinquency.

Indeed, their probability of membership in the chronic trajectory was

nearly zero. By contrast, the probability of transition to the chronic trajec-

tory from the high-opposition trajectory was 17 percent. Notwithstanding

these overall tendencies, the transition probabilities also make clear that

the childhood opposition trajectory is not even close to being a certain pre-

dictor of the subsequent trajectory of property delinquency. The entries in

Panel B make this clear. With the exception of the chronic trajectory, all

of the property trajectories were composed of large contingents of boys

from each of the opposition trajectories. Compared to simply correlating

the number of acts of property delinquency at each year from ages 13 to

17 with opposition at each age from 6 to 13, the dual model provides a far

richer, yet still comprehensible, summary of the relationships in the data.

The following statements were used to fit the dual model described in

this example:

proc traj data=combine out=b outstat=crimstat outplot=crimplot

outstat2=crimstat2 outplot2=crimplot2;

id id;

var opp1-opp5;

indep t1-t5;

model cnorm;

max 10;

ngroups 3;

order 2 2 2;

var2 del13-del17;

indep2 t5-t9;

model2 zip;

ngroups2 4;

order2 2 2 2 2;

run;





Extension 3: The Dual-Trajectory Model Generalized

to Include Predictors of Conditional Probabilities

We next demonstrate an extension that allows the conditional probabil-

ities linking trajectories across behaviors to vary as a function of individual-

level variables. The objective is to provide the statistical basis for analyz-

ing whether the conditional probabilities linking trajectories vary as a

function of individual-level characteristics and experiences. For example,

in the context of the childhood opposition/adolescent delinquency model,

experiencing a major life event at age 11 or 12 might alter the probability

of transition to the various delinquency trajectories. To illustrate this

extension, we extend the prior example with an analysis of how the prob-

abilities of transition from each of the childhood opposition trajectories to

each of adolescent property delinquency trajectories vary as a function of

the individual having experienced two potential turning-point events at

the time of the transition—not living with both parents at age 11 or 12,

hereafter referred to as ‘‘broken home 11-12,’’ and using drugs at age 12.

A full description of the technical details of how the basic dual-trajectory

model is extended to analyze such effects is described in Nagin (2005).

Here we provide only a brief outline. Let Y1 and Y2 denote the two longitu-

dinal series to be modeled in a dual-trajectory format, and let j and k index

the J and K trajectory groups for Y1 and Y2, respectively. In the basic dual

model, the direct products of model estimation are the parameters that spe-

cify the J and K trajectories of Y1 and Y2; the probability of membership in

each of Y1’s J trajectories, πj; and the conditional probability of member-

ship in Y2’s kth trajectory given membership in Y1’s jth trajectory, πk|j. To

ensure that πk|j is always between 0 and 1, this quantity is not directly esti-

mated. Instead, a set of parameters γ0k|j is estimated in which

πk|j = eγ0k|j,X

k

eγ0k|j j= 1; . . . ; J:

Let wi denote a vector of variables that are thought to be associated

with πk|j. The impact of these variables on πk|j is measured by a set of

parameter vectors γ 0k, whereby

πk|jðwiÞ= eγ0k|j+γ0

kwi

, Xk

eγ0k|jk+γ0

kwi j= 1; . . . ; J:

To avoid an unmanageable proliferation of parameters, the model

assumes that the effects of the variables included in wi do not depend

on the trajectory group membership for Y1. Under this assumption, the





various values of γ 0k|j are equal across the J trajectory groups in Y1; there-

fore, they can be denoted by γ 0k. Conceptually, this modification amounts

to assuming that the influence of a particular variable on the probability of

transition to a specific trajectory group k of Y2, as measured by γ 0k, does

not interact with trajectory membership for Y1.

Note that the no-interaction model continues to allow the intercepts,

γ 0k|j, to vary freely across Y1’s J trajectories. Consequently, the transition

probabilities, πk|j, will be different for two individuals with identical char-

acteristics wi but who had followed different trajectories for behavior Y1.

Thus, the no-interaction model still allows the trajectory that is followed

for Y1 to influence the probability of trajectory group membership for Y2,

even controlling for the variables in wi.

We turn now to our illustrative application of this model extension, in

which the conditional probabilities linking trajectories of opposition from

ages 6 to 13 with trajectories of property delinquency from ages 13 to 17

vary as a function of living in a broken home at age 11 or 12 and level of

drug use at age 12. Table 1 shows the γ 0k coefficient estimates with 95 per-

cent confidence intervals. For each trajectory group, the estimates should be

interpreted as the effect of its associated variable on the probability of tran-

sition to that delinquency trajectory relative to the low-delinquency trajec-

tory. The results indicate that controlling for the childhood opposition

trajectory, living in a broken home at age 11 or 12, has no effect on the tran-

sition probabilities, with the possible exception of those related to the

chronic trajectory. By contrast, drug use at age 12 has a positive and highly

significant effect on the probability of transition to the Low 2, rising, and

chronic trajectories relative to the Low 1 trajectory.

Table 1

The Impact of Living in a Broken Home and Drug Use on Trajectory

Transition Probabilities (Low 1 is the Comparison Group)

Delinquency Group and Variable Coefficient Z Score

Low 2

Broken home 11-12 0.114 0.48

Drug use 12 0.784 6.00

Rising

Broken home 11-12 0.123 0.47

Drug use 12 0.959 7.06

Chronic broken home 11-12 0.627 1.81

Drug use 12 1.37 9.23





Figure 4 provides a different perspective on these results. It reports

the probability of transition to the rising property delinquency trajectory

from each of the childhood opposition trajectories for two prototypical

individuals—one who does not use drugs at age 12 and one who is at the

75th percentile of the age 12 drug use distribution. Observe that regardless

of the childhood opposition trajectory group, drug use at age 12 greatly

increases the risk of transition to the rising delinquency trajectory group.

Extension 4: Multitrajectory Modeling

The dual-trajectory model is designed to measure the linkages between

the trajectories of two distinct but related outcomes. Conceptually, it is

straightforward to extend the dual model to more than two outcomes, but

as a practical matter, the addition of more outcomes results in an unma-

nageable proliferation of probability matrices linking the trajectories for

the various outcomes. For example, an extension to three behaviors

requires estimation of three matrices of joint probabilities—one for Out-

comes 1 to 2, another for Outcomes 1 to 3, and still another for Outcomes

2 to 3. An extension to four outcomes requires the estimation of six

matrices of joint probabilities. Still, there are many circumstances where

it would be valuable to link trajectories of three or more outcomes of

0

0.1

0.2

0.3

0.4

Lowopposition

Moderateopposition

Highopposition

Pro

bab

ility

No Drugs 75th Percentile Use

Figure 4

Probability of Transition From Each Childhood Opposition

Trajectory to the Rising Delinquency Trajectory





interest. The multitrajectory model is designed to provide this capacity in

a model of manageable size.

Figure 5 provides an illustrative application of the multitrajectory model

to three distinct but related forms of delinquency—property delinquency,

violent delinquency, and vandalism. As can be seen from the Figure 5, each

trajectory group is now defined not by one trajectory but by three trajec-

tories: one for property delinquency, a second for violent delinquency, and

a third for vandalism. As in the basic model, the size of the group is mea-

sured by the probability of group membership. As in the basic model, this

probability can be linked to characteristics of the individual. Also, as in the

base model, Proc Traj computes posterior probabilities of group member-

ship and related group membership assignments. Time-varying covariates

can also be added to the trajectories for each of behaviors, and the probabil-

ity of group membership can be related to characteristics of the individual.

This multitrajectory model is a form of the so-called constrained dual-

trajectory model described in Chapter 8 of Nagin (2005). The Poisson-

based form of the model is also laid out in Brame, Mulvey, and Piquero

(2001) and also applied in Piquero et al. (2002). The basic form of the

likelihood is

PðY1; Y2; . . . ; YKÞ=Xj

πj

Yk

fjk ðYkÞ;

where k is an index of the number of different outcome trajectories in each

trajectory group j. Note that f ð�Þ, which defines the distribution for each

such outcome by trajectory group, can be different across the outcomes. In

the example given above, the trajectories for each of the forms of delin-

quency were modeled using the Proc Traj Poisson model option. How-

ever, the use of the same distribution across outcomes is not required. For

example, one outcome might be modeled by the censored normal, another

by the logit option, and still another by the Poisson option.

The Proc Traj syntax for estimating the model shown in Figure 5 is

shown as follows:

PROC TRAJ DATA=MONTREAL OUTPLOT=OP OUTSTAT=OS

OUTPLOT2=OP2 OUTSTAT2=OS2 OUTPLOT3=OP3 OUTSTAT3=OS3;

VAR bat89-bat95; INDEP T1-T7; MODEL ZIP; ORDER 2 2 2 2 2;

VAR2 det89-det95; INDEP2 T1-T7; MODEL2 ZIP; ORDER2 2 2 2 2 2;

VAR3 vol89-vol95; INDEP3 T1-T7; MODEL3 ZIP; ORDER3 2 2 2 2 2;

MULTGROUPS 5;

RUN;





Figure 5

(A) Five-Group Multiple-Trajectory Model and

(B) Trajectory of Plots Organized by Behavior





Extension 5: Relating Trajectory Groups

to a Subsequent Outcome Variable

The dual model provides the capability to analyze the linkages

between the developmental trajectories of two related outcomes. Here we

demonstrate an option that links trajectory groups measured up to t peri-

ods for a specified outcome with another outcome variable, Oi, measured

only once on or after the termination of the trajectory. An example of the

type an analysis where this model would apply is the relationship of

trajectories of hyperactivity from ages 6 to 15 to performance on an aca-

demic achievement test at age 18. To accomplish this, the trajectory mix-

ture likelihood is modified to incorporate the probability of the observed

outcome, Oi:

PðYi;OiÞ = PðOiÞXj

πjPjðYiÞ:

We assume that the observed outcome depends only on the trajectory

groups. This dependency is modeled through a linear predictor (explained

below).

The form of PðOiÞ is selected to conform to the outcome data type.

For a count outcome, PðOiÞ can be specified according to the Poisson,

zero-inflated Poisson, or negative binomial distribution. For continuous

outcomes, PðOiÞ can be accommodated using the normal distribution.

Psychometric scale outcomes are accommodated by taking PðOiÞ to fol-

low a censored normal distribution. For binary data, PðOiÞ follows a bin-

ary logit distribution. The linear predictor, φ1πi1 +φ2πi2 + � � �+φJπiJ , is

used to model the relationship between posterior group membership and

outcome, where πi1; . . . ;πiJ are the posterior probabilities of group mem-

bership. For example, if PðOiÞ was specified to follow the Poisson distri-

bution, then lnðλÞ=φ1πi1 +φ2πi2 + � � �+φJπiJ , where λ is the expected

number of occurrences for the outcome of interest.

In the case of binary outcome data, it is assumed that PðOiÞ follows

the binary logit distribution:

PðOi = 1Þ= eφ1πi1 +φ2πi2 +��+φJπiJ

1+ eφ1πi1 +φ2πi2 +��+φJπiJ:

To illustrate, we investigate how the number of sexual partners at age

14 might differ by opposition trajectory groups in the childhood opposi-

tion model. The following statements fit the model with the outcome vari-

able described in this example:





proc traj data=a out=b outstat=crimstat outplot=crimplot

outcomeresults=crimresults;

id id;

var opp1-opp5;

indep t1-t5;

model cnorm;

max 10;

ngroups 3; order 0 2 2;

outcome nbp14;

omodel poisson;

run;

Table 2 gives the average number of sexual partners at age 14 with

95 percent confidence intervals by opposition trajectory group (ages 6-13).

These quantities were calculated from the φk coefficient estimates and

standard errors. The results show that the number of sexual partners differs

significantly by childhood opposition trajectory group, with greater levels

of oppositional behavior associated with higher numbers of sex partners.

The estimated average number of sexual partners in the low-opposition

group is 0.36 (95 percent CI: 0.31, 0.41). In contrast, the average number

of sexual partners in the moderate-opposition group is 0.67 (95 percent

CI: 0.46, 0.98), while the number of partners in the high-opposition group

is 1.62 (95 percent CI: 1.49, 1.75).

Extension 6: Calculating Group Membership Probabilities

as a Function of Time-Stable Covariates

Roeder, Lynch, and Nagin (1999) described an extension of the basic

model that allows group membership probabilities to vary as a function of

Table 2

Average Annual Number of Sexual Partners at Age 14 and 95 Percent

Confidence Intervals by Opposition Trajectory Group (Ages 6-13)

Opposition Trajectory

Group

Average Number

of Sexual Partners

95 Percent

Confidence Interval

Low 0.36 (0.31, 0.41)

Moderate 0.67 (0.46, 0.98)

High 1.62 (1.49, 1.75)





time-stable characteristics of the individual. Because such characteristics

act as predictors of trajectory group membership, they should be established

no later than the initial period of the trajectory. Denoting such predictors by

xi, the Roeder et al. model extension generalized equation (1) to follow a

multinomial logit function that is a function of xi:

πjðxiÞ= exiθj=XJ

1exiθj : ð5Þ

Figure 6 reports the results of an analysis in which the basic four-group

model described in the prior example is extended to allow trajectory group

probabilities to vary as a function of four classic childhood risk factors for

criminal and delinquent behavior in adolescence and beyond: low IQ, hav-

ing a parent with a criminal record, high risk-taking behavior, and poor

parenting. More specifically, the low-IQ variable distinguishes individuals

in the lower quartile of the IQ distribution. The high risk-taking variable

identifies individuals in the upper quartile of an index of risk taking. The

poor parenting variable distinguishes individuals whose parents were iden-

tified as having a poor or neglectful parenting style, and the criminal

parents’ variable identifies individuals with either a father or mother with

a criminal record.

Each coefficient estimate is associated with a particular risk factor for a

particular trajectory group. Each such estimate can be interpreted as the

log of the odds ratio of the impact of that risk factor on the probability of

membership in the specified group relative to the nonoffending group. In

general, each of the risk factors significantly distinguishes each of the

criminal trajectory groups from the nonoffender group.

An alternative way of representing the impact of one or more of these

risk factors is through calculations that directly represent their impact on

the probability of group membership. Figure 7 illustrates estimates of

group membership probabilities for the London model for six scenarios

about the level of the four predictor variables—low IQ, having a parent

with a criminal record, high risk taking, and poor parenting. Scenario 1

assumes that all are 0. This is equivalent to calculating group membership

probabilities for individuals with none of the above risk factors for delin-

quency. Scenarios 2 through 5 report these same probabilities for indivi-

duals with only one of the four risk factors included in the model. In

Scenario 6, the group membership probabilities are computed for indivi-

duals with all four of the delinquency risks.

The calculations illustrate the concept of cumulative risk (Rutter et al.

1975). Rutter et al. (1975) argued that in most circumstances, no single





risk factor is decisive in determining an individual’s vulnerability to

psychopathology but that the accumulation of such risks is decisive. The

calculations show that each risk factor increases the probability of mem-

bership in one or more of the delinquent trajectory groups, but no single

factor dramatically shifts the probabilities away from those in the no-risk

scenario. For example, consider Scenario 3. The model predicts that the

probability of membership in the rare group is .70 for individuals who

have at least one parent with a criminal record but who have none of the

other risk factors. The counterpart prediction for the high-chronic group

for these individuals is .039. In contrast, the predicted probabilities of

−2 −1 0

HC

LCAL

HC

LC

AL

HC

LC

AL

HC

LC

AL

LOG ODDS

1 2 3

Poor Parenting

High Risk - Taking

Criminal Parent

Low IQ

Figure 6

Log Odds With 95 Percent Confidence Intervals

for Risk Factors by Group

Note: HC= high chronic; LC= low chronic; AL= adolescent limited.





membership in the rare and high-chronic groups for individuals with no

risks are .89 and .006, respectively. Thus, the criminal parent risk factor

materially reduces the rare group probability and increases the high-

chronic group probability. Still, the basic ordering of the probabilities

remains the same—the rare group is much more likely than the high-

chronic group. However, the presence of all four risks results in a dramatic

shift. The probability of membership in the high-chronic group increases

from nearly 0 in the no-risk scenario to .48 in the all four risk scenarios.

The calculations that form the basis for entries in Figure 7 are based on

the logit model coefficient estimates illustrated in Figure 6. Proc Traj, in

combination with standard SAS data-handling features, can be used to

perform these calculations with the following set of commands:

Rare Adolescent Limited

Low Chronic High Chronic

No

Ris

ks

Low IQ

Crim

inal

Par

ent

Hig

hR

isk-

Tak

ing

Poo

rP

aren

ting

All

Fou

rR

isks

Figure 7

Predicted Group Membership Probabilities: London Data





data predict;

input lowiq pbeh daring crimpar;

cards;

0 0 0 0

1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1

1 1 1 1

run;

data combine;

set predict london_data;

run;

proc traj data=combine out=outdata outstat=crimstat outplot=crimplot;

var y1-y11;

indep t1-t11;

model zip;

ngroups 4;

start -4.57 -7 5.96 -1.38 -20.5 25.4 -8.0 -4.37 5.55 -1.52

-1.7 0 0 0 0 -1.7 0 0 0 0 -2.5 0 0 0 0;

order 0 2 2 2;

iorder -1 -1 -1 -1;

risk lowiq crimpar daring pbeh;

run;

proc print data=b (obs=6);

run;

The first two data steps preceding the call to Proc Traj append six

dummy records to the top of the estimation data set. In the six dummy

records, the values of the four risk variables are set to correspond to the

six scenarios for which we want to calculate group membership variables.

However, because the number of convictions in each of the periods from

ages 10 to 32 is left unspecified in the dummy records, SAS specifies their

values as ‘‘missing.’’ As a consequence, these records have no impact on

the estimated model coefficients, their standard errors, or the calculation

of various model fit statistics, such as the Bayesian information criterion.

Thus, they are referred to as dummy records.

The final processing step involves printing out the first six records of

a data set called ‘‘outdata.’’ This data set is produced by Proc Traj. It

includes the posterior probability of membership in each trajectory group





for each case in the estimation data set. The posterior probability of mem-

bership in group j for case i is based on application of Bayes’s theorem.

For each case i, group j’s posterior probability is calculated using the esti-

mated model coefficients and the data for that case. For real cases with

actual conviction data, this calculation takes into account the history of

conviction. For example, an individual whose convictions are limited to

his or her adolescence will have a high probability of membership in the

adolescent-limited group. However, for the dummy cases, in which the

conviction history is deliberately specified as missing, the calculation

reduces to the desired quantity—that specified by equation (5).

Extension 7: Wald Tests of the Equality

of Coefficient Estimates

The two primary outputs of model estimation are the model’s para-

meter estimates and their variances and covariances. The latter form the

basis for the test statistics reported in the output of Proc Traj of whether

the coefficient estimates are significantly different from zero. For exam-

ple, the confidence intervals showing whether the various risk factors were

significant predictors of membership in one of the criminal trajectories

compared to the nonoffender trajectory, given in Figure 6, were easily

derived directly from the Proc Traj output.

However, many other interesting and important statistical tests can be

conducted based on the variance-covariance estimates of the parameter

estimates. One is whether the various risk factor coefficient estimates dif-

fer across the three criminal trajectory groups. Differences in their mag-

nitude would imply that they differentially predict membership in the

various criminal trajectory groups. Alternatively, if they are individually

significantly different than zero but the null hypothesis of their equality

cannot be rejected, this would imply that while the risk factor is a signifi-

cant predictor of criminality, it does not predict among its alternative

developmental courses.

As elaborated in Nagin (2005), such hypotheses can be tested with the

Wald test. The Wald test is a chi-square-based test with degrees of freedom

(df ) equal to the number of equality constraints being tested. A newly

developed SAS macro called trajtest can be used to conduct Wald tests of

the equality of one or more coefficients. For example, a test of the equality

of the low IQ coefficients across the three criminal trajectory groups yielded

a Wald-based chi-squared statistic of .98 with 2 df , which is far short of





significance. This result implies that low IQ is not a significant predictor of

which of the criminal trajectories an individual might follow. Thus, while

the results in Figure 6 showed that low IQ was a risk factor for criminality,

the chi-square-based statistic implies that low IQ does not distinguish

among the specific developmental courses of criminality. On the other hand,

the companion test for the risk-taking variable was significant at the .05

level. This implies that risk taking was not only a significant predictor of

criminality but also of its specific trajectory.

The Wald test can also be used to test whether trajectories are distinc-

tive in the sense that they are not parallel. Two or more trajectories would

be considered parallel if the constant terms of the trajectories (i.e.,

βj0Þwere significantly different but the coefficients of higher order terms of

the polynomials describing the trajectories of the respective groups (i.e.

βj1;β

j2; . . .Þwere not. We applied this testing procedure to compare the

low- and high-chronic trajectories. We found that both their intercepts and

their linear and quadratic terms were significantly different according to

the Wald test. The chi-square statistic for the intercept contrast is 15.71

(df = 1), and the companion chi-square test statistic for the equality of the

linear and quadratic terms was 16.44 (df= 2).

The syntax of the calls to the trajtest macro for conducting the above

reported tests was as follows:

%TRAJTEST (’lowiq2=lowiq3=lowiq4’)

/* low iq equality test */

%TRAJTEST (’daring2=daring3=daring4’)

/* risk taking equality test */

%TRAJTEST (’interc3=interc4’)

/* intercept equality test */

%TRAJTEST (’linear3=linear4,quadra3=quadra4’)

/* linear & quadratic equality test */

Extension 8: Plotting the Impact of Time-Varying Covariates

Jones et al. (2001) demonstrated the capability of adding time-varying

covariates beyond age or time into the specification of the trajectory. As

developed at length in Nagin et al. (2003) and Nagin (2005), the objective

of this modeling extension is to analyze the effect of turning-point events,

such as divorce, or therapeutic interventions, such as counseling, on the

behavioral trajectories. (We use the term effect advisedly in full recogni-

tion of the difficulties of causal interpretation in observational data. See





Haviland and Nagin [2005] and Nagin [2005] for a discussion of causal

inference in the context of group-based trajectory modeling.)

Here we demonstrate a new feature that aids in the graphical presenta-

tion of the estimated effect of the covariate of interest on each trajectory.

Specifically, the function called plottcov directs Proc Traj to calculate the

trajectory for each group under a user-specified set of values of the covari-

ate of interest. These calculations are done postmodel estimation based on

the estimated values of the coefficients that define the trajectory over time.

These include the coefficients measuring the effects of all the covariates

included in the trajectory.

We demonstrate this capability with a group-based trajectory analysis

of the effect of gang membership on violent delinquency. This analysis is

based on self-reports from the Montreal data on violent delinquency from

ages 11 to 17 and companion self-reports of whether or not the boy was

involved in a delinquent group at that age. With the addition of gang

membership as a time-varying covariate, the form of the trajectory for

each group was as follows:

lnðλjt Þ=βj0 +β

j1Ageit +β

j2Age

2it +α

j1gangit;

where gangit is a dichotomous indicator variable that equals 1 if subject

i reports being in a gang in period t. We model self-reported frequency of

violent delinquency as a Poisson random variable, and thus each trajectory

is defined by the Poisson rate parameter, λjt , over time.

A five-group model was estimated. Note that the model allows the

effect of the gang membership status variable to vary freely across trajec-

tory groups, which allows for the possibility of differential effects. In this

analysis, the estimate of αj1 was positive and highly significant for each

group, which implies that gang membership is associated with increased

violence.

Figure 8 is a graphical depiction of the plottcov-generated calculations

for two of the five trajectories. For each trajectory, the graph compares the

predicted trajectory for an individual who stays out of a gang from ages

11 to 17 with that of an individual who stays out of gangs from ages 11 to

13 but at age 14 joins a gang and remains a gang member thereafter. As

indicated, the gang effect was positive and statistically significant for all

groups, but the figure shows that the effect size was larger for what we call

the moderate-chronic group than for what we call the low group.

The following commands were used to create the predicted trajectories

shown in Figure 8:





proc traj data=a out=b outstat=viostat outplot=vioplot1;

var vio11-vio17;

indep age11-age17;

model zip;

tcov gang11-gang17;

plottcov 0 0 0 0 0 0 0;

ngroups 5;

order 2 2 2 2 2;

run;

proc print data=vioplot1;

run;

proc traj data=a out=b outstat=viostat outplot=vioplot2;

var vio11-vio17;

indep age11-age17;

model zip;

tcov gang11-gang17;

plottcov 0 0 0 1 1 1 1;

0

1

2

3

4

5

11 12 13 14 15 16 17Age

Pre

dic

ted

Rat

e

Low—Never Joins a Gang Low—Joins a Gang at 14 Chronic—Never Joins a Gang Chronic—Joins a Gang at 14

Figure 8

Illustration of Estimated Effect of Gang Membership on Violent

Delinquency for the Low 2 and Moderate Chronic Trajectories





ngroups 5;

order 2 2 2 2 2;

run;

proc print data=vioplot2;

run;

The key commands are in bold. The command ‘‘tcov gang11-gang17’’

adds the gang membership status variable to the trajectory. Note that addi-

tional covariates beyond gang membership status at each age could have

been included in the tcov command. The command ‘‘plottcov 0 0 0 0 0

0 0’’ directs Proc Traj to calculate the predicted trajectories postestimation

under the assumption that the gang membership status variable equals 0 in

all periods. This command produced the predicted trajectories under the

non-gang-joining condition shown in Figure 8. The predicted trajectories

were stored in the file named ‘‘vioplot1’’ and were printed out using

SAS’s standard proc print function. To obtain the predicted trajectories for

the condition where the individual joins a gang at age 14 (i.e., plottcov 0 0

0 1 1 1 1), the entire process has to be repeated. Even though the plottcov

calculations were performed postestimation, technical obstacles make it

impossible to include more than one use of plottcov in a call to Proc Traj.

Thus, the entire model must be reestimated to do the second set of calcula-

tions that were required to produce Figure 8.

The figure itself was created by entering the predicted trajectories in

vioplot1 and vioplot2 into a stand-alone graphing software package.

Also, note that the two vioplot files also included the actual trajectories

of violent delinquency for each trajectory group. For each trajectory

group, the calculation of the actual trajectory is based on the actual

values of violent delinquency for each individual weighted by their pos-

terior probability of membership in that trajectory. Because actual gang

membership experience has a pronounced positive association with vio-

lent delinquency, the actual trajectories are no longer comparable to the

predicted trajectories under the hypothetical set of conditions specified

in the plottcov command.

Extension 9: Sample Weights and Exposure Time

The final example illustrates the capacity to incorporate sample weights

and adjust for exposure time in the ZIP model. These options will be

shown using data from the Rochester Youth Development Study.





The Rochester Youth Development Study tracked a sample of students

who were in the seventh and eighth grades in Rochester, New York, public

schools for the spring semester of the 1988 school year. Males and

students from high-crime areas were more heavily sampled since it was

assumed that they were at greater risk for offending. The project was a

12-wave prospective study where members of the sample and one of their

parents were interviewed semiannually from 1988 to 1992 and annually

from 1994 to 1996.

Due to the oversampling of individuals from high-crime areas, the

data for study participants must be weighted in model estimation to

obtain a model that provides unbiased estimates of the representation of

each trajectory group in the population and also to obtain valid standard

errors. A robust estimator of the variance-covariance matrix for the para-

meters from the weighted likelihood function is obtained using a sand-

wich estimator.

In addition, because the time between assessment periods was not

constant over the course of the study and across study participants, it is

imperative that we account for differences across individuals and over

time in the participant’s ‘‘exposure time’’ to commit crimes between

assessments. For example, in the early stages of the study when assess-

ments were made semiannually, counts of self-reported delinquent acts

referred to approximately 6 months of exposure time, whereas later in the

study, the counts referred to approximately 12 months of exposure time.

More generally, it is critical that time between assessments be taken into

account in any analysis of event frequency.

An adjustment for exposure time is also required if individuals are

somehow placed in a situation where they are restricted from engaging

in the activity of interest. For example, the exposure time adjustment

was first demonstrated in the Piquero et al. (2001) analysis of the arrest

histories of individuals who had been under the supervision of the Cali-

fornia Youth Authority. The purpose of the exposure time adjustment

was to account for spells of imprisonment, during which times indivi-

duals cannot be arrested for crimes that would appear on their arrest

record.

The use of exposure variables modifies the Poisson rate equation as

follows:

lnðλjt Þ=βj0 +β

j1 Ageit∗Expositð Þ+β

j2 Ageit∗Expositð Þ2:

The following statements fit a two-group model to the arrest counts:





proc traj data=a out=b outstat=crimstat outplot=crimplot;

id id;

weight wt;

var ac1-ac12;

indep t1-t12;

expos e1-e12;

model zip;

ngroups 2; order 2 2; iorder 0 2;

run;

Figure 9 shows the estimates for the two trajectories of λjt . The first

group is composed of individuals with a very low rate of offending over

the 12-wave period. The size of this group is estimated at 78.1 percent.

The other trajectory group (21.9 percent) shows very high and increasing

levels of offending during the 9th through 12th waves.

13 14 15 16 17 18 19 20 21

0

10

20

30

40

AGE

An

nu

al A

rres

t R

ate

21.9 percent 78.1 percent

Figure 9

Annual Arrest Rate Versus Age





Conclusion

The purpose of this article was to describe extensions of a SAS-based

procedure for estimating group-based trajectory models that have been

developed since the appearance of Jones et al. (2001). These include the

following:

1. reporting confidence intervals on trajectories and group membership

probabilities,

2. calculating group membership probabilities as a function of time-stable

covariates,

3. conducting Wald tests of the equality of coefficient estimates,

4. plotting the impact of time-varying covariates,

5. estimating dual-trajectory models with the ability to include predictors of

the conditional probabilities that link trajectories of two different outcomes,

6. relating trajectory groups to a subsequent outcome variable,

7. including sample weights and adjusting for exposure time in the ZIP model,

and

8. linking trajectories of three or more outcomes of interest with the multitra-

jectory model.

It our hope that these expanded capabilities will increase the use of

group-based trajectory modeling in appropriate problem domains invol-

ving the analysis of longitudinal data.

References

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Haviland, A. and D. S. Nagin. 2005. ‘‘Causal Inference With Group-Based Trajectory

Models.’’ Working paper, Carnegie Mellon University.





Jones, B. L., D. Nagin, and K. Roeder. 2001. ‘‘A SAS Procedure Based on Mixture Models for

Estimating Developmental Trajectories.’’ Sociological Methods & Research 29:374-93.

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Nagin, D. 1999. ‘‘Analyzing Developmental Trajectories: Semi-Parametric, Group-Based

Approach.’’ Psychological Methods 4:39-177.

———. 2005. Group-Based Modeling of Development. Cambridge, MA: Harvard University

Press.

Nagin, D., L. Pagani, R. Tremblay, and F. Vitaro. 2003. ‘‘Life Course Turning Points: A Case

Study of the Effect of School Failure on Interpersonal Violence.’’ Development and Psy-

chopathology 15:343-61.

Nagin, D. S. and R. E. Tremblay. 2001. ‘‘Analyzing Developmental Trajectories of Distinct

but Related Behaviors: A Group-Based Method.’’ Psychological Methods 6 (1): 18-34.

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From Longitudinal Studies?’’ Paper presented at the National Institute of Justice,

Washington, DC.

Piquero, A., A. Blumstein, R. Brame, R. Haapanen, E. Mulvey, and D. S. Nagin. 2001.

‘‘Assessing the Impact of Exposure Time and Incapacitation on Longitudinal Trajectories

of Offending.’’ Journal of Adolescent Research 16:54-76.

Piquero, A. R., R. Brame, P. Mazzerole, and R. Haaplanen. 2002. ‘‘Crime in Emerging

Adulthood.’’ Criminology 40:137-70.

Roeder, K., K. Lynch, and D. S. Nagin. 1999. ‘‘Modeling Uncertainty in Latent Class Mem-

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Bobby L. Jones is a principal research analyst and statistician, Heinz School of Public Policy

and Management, Carnegie Mellon University. Recent publications include ‘‘Mixture Models

for Linkage Analysis of Affected Sibling Pairs and Covariates’’ (with Bernie Devlin, Silviu

Bacanu, and Kathryn Roeder) in Genetic Epidemiology (2002) and ‘‘Underdevelopment of

the Postural Control System in Autism’’ (with Nancy Minshew, Kim Sung, and Joseph

Furman) in Neurology (2004).

Daniel S. Nagin is the Teresa and H. John Heinz III Professor of Public Policy and Statistics,

Heinz School of Public Policy and Management, Carnegie Mellon University. His research

focuses on the evolution of criminal and antisocial behaviors during the life course, the deter-

rent effect of criminal and noncriminal penalties on illegal behaviors, and the development of

statistical methods for analyzing longitudinal data. He is the author of Group-Based Model-

ing of Development (2005).





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